Well, I'm not an MP. But, I know a lot about them just from being in the Air Force.
There is an extremely good chance of deploying. The AF police go for 6 months at a time and usually have a 12 month break in between deployments. But, the AF also goes to better deployments. If you chose Army, Iraq is almost a given. I'd chose the AF police honestly.
You just have to take the ASVAB test. The scoring for being an MP is pretty low. You pretty much have to answer your name correctly. As for the background, just tell them you haven't done anything as long as you have a clean record. Chances are you are only going to get a Secret security clearance like me. And tell them you rarely drink. Trust me on that one.
1. The pay goes by rank. So for example, you would get paid the same as someone the same rank that hands out towels at the gym. But, you'll get hazard duty pay and tax free when you deploy. Which I disagree about that though. I think a MP should get even more when deployed to a hostile environment. I got the same hazard duty pay in a peaceful deployment as an Army ranger on the front line in a hostile place for example. But, I'm an E-5 (Staff Sargeant) and my base pay is $2400/month, $360/month for food, and $1840/month for housing (housing is different depending on the area). It is definitely liveable pay.
2. I would think that it would help in becoming a civilian police officer. But, I would honestly go straight to trying to be a civilian P/O instead of joining the military if that's all your looking to get out of it.
3. I'd say about 70% of the MP's here at my base hate being MP's and wouldn't do it again if that helps. But, then again, a lot of them love the Authority they have. So, to each their own. I'm an Aircraft Mechanic and my job is pretty cool. It's better than being an MP for sure.
4. I'd say the deployments and the hours and bullshit that they have to deal with.
If you have any other questions, I'll try to help with them.